On the other hand, given a hereditary torsion theory \(\tau =(\mathcal {T},\mathcal {F})\)  one can construct a Giraud subcategory of \(\mathcal {G}\)  as follows. We say that an object \(X\in \mathcal {G}\)  is \(\tau \) -local if both \(X\in \mathcal {F}\)  and \(E(X)/X\in \mathcal {F}\) . The full subcategory of \(\mathcal {G}\)  of all the \(\tau \) -local objects is denoted by \(\mathcal {G}/\mathcal {T}\)  and the inclusion functor by \(\mathbf {S}_{\tau }\colon \mathcal {G}/\mathcal {T}\rightarrow \mathcal {G}\) ; one can give an explicit construction of the left adjoint \(\mathbf {Q}_\tau \colon \mathcal {G}\rightarrow \mathcal {G}/\mathcal {T}\)  (called the \(\tau \) -quotient functor) to \(\mathbf {S}_\tau \) , and prove that this functor is exact. Let us remark that we adopt the notation “\(\mathcal {G}/\mathcal {T}\) ” because this category is equivalent to the Gabriel quotient of \(\mathcal {G}\)  over \(\mathcal {T}\)  (for more details on these constructions see [1]}, [2]} or [3]}).
